Modeling of a Field-Modulated Permanent-Magnet Machine

[Pages:16]Modeling of a Field-Modulated Permanent-Magnet Machine

Authors:

Xianglin Li, K. T. Chau, Yubin Wang

Date Submitted: 2019-02-27

Keywords: finite element analysis (FEA), Modelling, field-modulated permanent magnet (FMPM) machine, d-q frame

Abstract:

In this work, an effective field-modulated permanent-magnet (FMPM) machine was investigated, in which the spoke-magnet outer rotor and open-slot stator were employed. The objective of this paper is to provide the mathematical modeling analysis that was performed for the purpose of control research on this type of FMPM machine. The simulation results by means of finite element analysis (FEA) are given to verify the theoretical analysis and the validity of mathematical model. A prototype machine was also fabricated for experimentation. Both the analytical model and the FEA results are validated by experimental tests on the prototype machine.

Record Type: Published Article

Submitted To: LAPSE (Living Archive for Process Systems Engineering)

Citation (overall record, always the latest version): Citation (this specific file, latest version): Citation (this specific file, this version):

LAPSE:2019.0393 LAPSE:2019.0393-1 LAPSE:2019.0393-1v1

DOI of Published Version:

License: Creative Commons Attribution 4.0 International (CC BY 4.0)

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energies

Article

Modeling of a Field-Modulated Permanent-Magnet Machine

Xianglin Li 1,*, K. T. Chau 2 and Yubin Wang 1 1 College of Information and Control Engineering, China University of Petroleum, Qingdao 266580, China; yubwang5190@ 2 Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, China; ktchau@eee.hku.hk * Correspondence: xianglinli@upc.; Tel.: +86-15066851211

Academic Editor: Joeri Van Mierlo Received: 4 August 2016; Accepted: 12 December 2016; Published: 19 December 2016

Abstract: In this work, an effective field-modulated permanent-magnet (FMPM) machine was investigated, in which the spoke-magnet outer rotor and open-slot stator were employed. The objective of this paper is to provide the mathematical modeling analysis that was performed for the purpose of control research on this type of FMPM machine. The simulation results by means of finite element analysis (FEA) are given to verify the theoretical analysis and the validity of mathematical model. A prototype machine was also fabricated for experimentation. Both the analytical model and the FEA results are validated by experimental tests on the prototype machine.

Keywords: field-modulated permanent magnet (FMPM) machine; d-q frame; modeling; finite element analysis (FEA)

1. Introduction

The operation of field-modulated permanent magnet (FMPM) machines relies on the "magnetic gearing effect" resulting from the magnetic field modulation [1], which can be derived from the coaxial magnetic gear by replacing the gear's high-speed rotor with a stationary armature fed by symmetrical three-phase winding currents [2]. It has been discussed that FMPM machines can develop a high torque density by using the high-speed armature field operation with low armature pole-pairs and slot number while the rotor still rotates at a low speed to transmit a high torque [3?5]. Due to the "magnetic gearing effect" coupled with its compact structure, high torque density, and high efficiency, the FMPM machine is promising for low-speed direct-drive applications such as wind power generation [6], wave energy conversion [7], and electric vehicles [8]. Recently, in order to verify the potential of FMPM machines for industry applications, many improved FMPM topologies have been successively proposed and analyzed [9,10]. Furthermore, in order to improve the power factor, a dual-stator spoke-magnet FMPM machine was developed and its attractive characteristics were demonstrated in [11].

The work of foregoing research on FMPM machines mainly focuses on the analysis of electromagnetic characteristics, such as achieving a high torque or force density, core loss calculation, magnetic circuit optimization, among others [12]. These analysis results have certainly confirmed the advantageous features of FMPM machines on torque capability and efficiency compared to traditional permanent magnet (PM) machines [13]. However, so far, a detailed modeling analysis of these FMPM machines prepared for the purpose of driving control has not been reported. It is known that the direct torque control or field-oriented control based on a synchronous d-q frame are commonly used for PM motor drive [14]. However, due to the "magnetic gearing effect", the rotational velocities of the PM rotor and armature field in an FMPM machine are different, thus it is necessary to discuss

Energies 2016, 9, 1078; doi:10.3390/en9121078

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Energies 2016, 9, 1078

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how to define the synchronous d-q frame. Meanwhile, as a foundation for its driving control system, the mEantehrgeiems 2a01t6ic, 9a,l10m78odeling of FMPM machines based on the newly defined synchronous d2 -oqf 1f5rame should be established.

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NNuummbbeerr ooff ppoollee--ppaaiirrss

((bb))

FigFFuiiggrueurr3ee. 33N.. NoN-oolo--llaoodaaddaiaari-irrg--aggpaappflffulluuxxxdddeenennssistiiyttyyooofffttthhheeeppprrrooopppooossseeeddd 111888---ssslllooottt///888--pp-pooolleeleFFFMMMPPMPMMmmmaaccahhciihnnieen..e((a.a))(aWW) Waavvaeevffooerrfmmor;;m; anaadnn(ddb()(bbh))ahhraamrrmmonooinnciicscpsseppceetccrtturruummm. ..

3. Mathematical Modeling 3. Mathematical Modeling

In the proposed machine, it can realize magnetic field modulation between rotor 14 pole-pairs andInstthaetopr r4oppoosleed-pmairasc,hainned, ibtoctahnmreaaglnizeeticmfaigelndestihcafiveelddimffeordeunltatmioenchbaentiwcaelernortaottionrg1v4eploocliet-iepsa,irs anndasmtaetloyr, 4acphoielev-ipnagirtsh,ean"dmbagonthetmicaggenaertiincgfieeflfdescth"a. vHeodwifefveerer,nttamkiencghathneicaplorloetraatitniogGvrer lionctoitiaecsc, onuanmt,ely, achbioetvhinmgatghnee"timc faigelndestihcagveeatrhinegsaemffeecet"le.cHtroicwaleavnegr,utlaakrinveglothcietypoleee. AractcioorGdriningttooatchceo4unpto,lbeo-pthaimr faieglnde, tic fietlhdes healevcetrtihceasnagmleebeeletwctereicnalaadnjagcuelnatr svleoltocEiMtyFvee. cAtocrcsorids i8n0g?;tootnheth4epootlhee-rpahiarnfide,ldb,atsheedeolencttrhice a1n4gle betpwoleee-npaaidr jfaiecledn,titslios t2E80M?.FTvheacttios,rswihse8t0he;roitnisthpeloottthederbhaasendd,obnaase4dpoonle-tphaeir14orp1o4lep-oplaei-rpfiaierldfi,eiltdi,sth2e80. Thsaatmise, wslhoet tEhMerFitsitsapr ldoitategdrabmasceadnobneao4bptaoinlee-dpaaisr oshr o1w4 pnoilne-Fpiagiurrfiee4lda,,bt,hreessapmecetisvleoltyE. MHeFnsctea,ridt ialgsroam canindbeicaotbetsatinhaetdtahse s"hmoawgneitnicFgigeaurrieng4ae,fbfe, crte"spcaenctimvealkye. tHheen4cep,oilte-aplasoir iannddic1a4tepsotlhea-pt atihrem"amgangetnicetic

gefaireilndgs etfofgeecth"ecrancomntarkibeuttheet4o ptohlee-penaeirrgayndco1n4vpeorsleio-np,aitrhmusagancehtiiecvfiinegldssigtongifeicthanetrlycohnitgrihbutoterqtuoethe

enceargpyabciolintyveinrstihoen,ptrhoupsoasecdhiFevMinPgMsimgnacifihcinaen.tly high torque capability in the proposed FMPM machine.

114 ,4 ,55 7 ,7 ,1166

77,1,166 1144, , 55

EEbb

bb 1177,, 88

SSlloott nnuummbbeerr 1133,, 44 1188,, 99

1122, , 33

qq 2222..55oo dd

ff--44==GGrrrr aa

22,, 1111 8800oo

66,, 1155

2222..55oo EEaa

EEcc cc

1188,, 99

SSlloott nnuummbbeerr 1133,, 44

228800oo

1177,, 88

1 ,1 ,1100

66,, 1155 22,,1111

66..4433ooqq dd

aa

ff--1144==rr

66..4433oo EEaa

112 ,2 ,33

11, , 1100

SSlloott EEMMFF

SSlloott EEMMFF

EEcc

cc

((aa))

EEbb bb ((bb))

FigF1F1u88iigg--rssueullorore4ett//.8844--..ppSoSloSolllloeoetttFFeMMeleelleePcPcctMMtrtrrooommmmmaaoooccttthhiiivvviinneeeee.f.ffooo((aarrr)cc)ceeeBBaa(((ssEEEeeMMMddFFoFo))nn) ss4s4tttaapparrorollddeedi-i-iaappaggaagrriirararammffmiieellaaddann;;nddaadnndddd-d-qq(-(bqbaa))xxabbeexaassessseeddddeedffooeiinnnnfiiin1t1tii44iootninppooonoolleeff-o-ppttfhhaaetieirhr pfepfiirerepoollddppr.o.oopsseeoddsed 18-slot/8-pole FMPM machine. (a) Based on 4 pole-pair field; and (b) based on 14 pole-pair field.

Energies 2016, 9, 1078 Energies 2016, 9, 1078

5 of 15 5 of 15

To Treoarliezaelizthe ethteratnrasnfosfromrmataitoionnfrfroommssttaattoor refeerreenncceeffrraammeetotorortootrorerferfenrecne cferafmraem, teh,etdh-eq da-xqesaxes

of tohfethperopproopseodsedmmacahcihnineeccaannbbee ddeeffiinneeddaaccccoordrdininggtotroortaottiantgin4gp4olpe-oplaei-rpfaieirldfioerld14opro1le4-ppaoilref-ipeladirasfield

as sshhoowwnn iinn FFiigguurree44aa,b,b, ,rersepsepceticvteivlye,lyin, iwn hwichhichf-4afn-4danfd-14 ref -p14rerseepntretsheenmt tehcheamniecaclhaanngicleasl aonf gale4s of a 4 ppoollee--ppaairirfifieeldldaannddaa1144ppoolel-ep-paiarirfifieledldrerlealtaivtievetotothteheiniintiiatliaplopsiotsioitni,onre,srpeescptievcetliyv.eAlys. Ashsoswhnowinn in FiguFirgeu2rae,2sat,asttoartoar-aax-aisxiissisdedfiefninededaalolonnggtthhee cceennttrraall aaxxiissooffpphhasaeseAAatatthtehieniitniaitliaplopsiotisointiownhewrehethree the PtoMbPbreefleMluaccotxfoilnounlnsixsnisisslhkittnieaepknngaitetsgwwogefoiiottvphfhehpraaahn--saaeaexsdxeiiAsisAn,, ratarehnenaedadccphhtrtheeohesspettoqhhq-s-aeeeaxdpxpisomiossisiasiticsitvh9iv9e0in0e?meme:aelxealciexmtcirtmiurcmiauclamvdlaedvlguearegle.ureTeeshe.aesThnahe,hatehdenea,oditnfhoiedtfi-aiadnlx-diiats-ixa.axilTsi.shd-eiTsahxfcoeihslolfoisoswelnlcionhtwgoosineng relationship is governed in the proposed machine:

Gefr-==1G4epfr pp=1rrs4ppprrrsrrr

1

=

GrG1r f

4

f

-4

(1) (1)

whewrheerer anr dandeaeraerethtehemmeecchhaanniiccaal andd eelleecctrtricicaal lanagnlgelseosfothf ethreotroortpoorspitoiosnit,iorens,preecstipveeclyti;vper layn;dprpsand

ps aarreetthhee ppole-ppaaiirrnnuummbbererofofrortootroarnadnsdtasttoart,orre,srpeescptievcetliyv.eIlty.caInt cbaenfobuenfdouthnadt wthitaht twheithrothateiornotoaftion of tthheerroottoorr,,ththeemmeecchhaanniiccaall aanngglleess ff--4 4anadndf-1f4-a14reardeifdfeirffeenrtednut edutoe tthoeth"em"amgnaegtincegtiecagrienagrienfgfecetf"f.ect". HowHeovweerv, tehr,eitrheelirecetrleiccatrlicaanlgalensgalerse athree sthaemseaamseea.sThea. tTish,atthies,d-tqheaxde-sq daexfiesnidtieofninbitaiosendboanse4dpoonle4-pair fieldpooler-p14airpfoieleld-poarir14fipeoldle-ips aiinr ffiaecldt eisqiunivfaaclteenqtuiifviatleisnttrifeaittiesdtrferaotmed tfhroemvitehwe vpioeiwnpt ooifnteloefcetlreicctarlicaanl gle. Henacneg,leto. Hreednucec,e tcoonrefudsuicoencaonndfussiimonplainfyd asnimalpylsiifsy, tahnealfyoslilso,wthinegfomllaotwhienmg amticaathlemmoadtiecal lismeostdaeblliisshed baseesdtaobnlisthheedrobtaosredeloencttrhicearloatonrgelelecter.ical angle e.

3.1.3M.1.aMthaetmheamticaatilcaMl Modoedl eilninStSattaotrorRRefeeferreenncceeFFrraammee

As shown in Figure 5a, the three-phase PM flux linkage of the proposed machine can be exApsresssheodwans: in Figure 5a, the three-phase PM flux linkage of the proposed machine can be expressed as:

pma

=pma m

mccooss (e e

)

pmb pmc

=pmbmmccooss (e e-1201o20o) =pmc m mccoos(e e+1201o20o )

(2) (2)

whewreheremism tishethpeepaekakvavlaulueeoof fththeepphhaassee PPM fflluxx lliinnkkaaggee. .

Flux-linkage (Wb) Flux-linkage (Wb)

1.5

1.3

1

1.2

0.5

1.1

0

pma -0.5

pmb

pmc

0.1

-1

0

pmd

pmq

pm0

-1.5 0

60

120 180 240 300 360

Rotor position, e (o)

(a)

-0.1 0

60 120 180 240 300 360 Rotor position, e (o)

(b)

FiguFirgeu5r.eP5M. PMflufxlulxinlkinakgaegeofotfhtheepproroppoosseeddmmaacchhiinnee iinnddiiffffeerreennt trerfeefreernecnecferafrmaems:e(sa:)(sat)astotartroerferreefnecreence framfrea;m(be;)(db-)qdr-eqfreerfeenrecnecferafrmame.e.

Then, the total phase flux linkage excited by PMs and phase current on load can be expressed as:

Then, the total phase flux linkage excited by PMs and phase current on load can be expressed as:

a b

=

a

Lab a Mcba

Laa

MMbaab

MLcabb

Mab Mac ia

LbbMaMc bc ib

M Mcb Lcc

bc

ic

iibapppmmmabc+

pma pmb

(3) (3)

where ia, ib, and ic are thepchase curreMntcsa, LaMa, Lcbbb, anLdcc Lcc areitche phaseseplmf-cinductances as shown in

Figure 6a, and Mab, Mac, Mba, Mbc, Mca, and Mcb are the phase mutual inductances as shown in Figure 6b. where ia, ib, and ic are the phase currents, Laa, Lbb, and Lcc are the phase self-inductances as shown

in Figure 6a, and Mab, Mac, Mba, Mbc, Mca, and Mcb are the phase mutual inductances as shown in

Figure 6b.

Energies 2016, 9, 1078 Energies 2016, 9, 1078

6 of 15 6 of 15

Self-Inductance (mH) Mutual-Inductance (mH)

40

Lcc

Laa

Lbb

0

Mbc

Mca

Mab

-3 30

-6 20

-9

10

-12

0

0

60

120 180 240 300 360

Rotor position, e (o)

(a)

-15 0

60

120 180 240 300 360

Rotor position, e (o)

(b)

FigFuigruer6e. 6S.eSlfe-lfa-nadndmmutuutuala-li-nindduuccttaannccee ooff tthhee pprrooppoosseedd mmaacchhininee. .(a(a) )SSelefl-fi-nidnudcutcatnacnec; ea;nadn(db)(bm)umtuuatlual indinudcutacntacnec. e.

Table 1 lists the characteristic information of inductance. It can be seen that the first and second harTmabolneic1colimstpsotnheentcshoafrsaecltfe-irnisdtuicctainnfcoermareatsiiognniofifcainntd, uwchtaicnhcaec. cIotucnatnfobr earsoeuennd t1h0a.1t%thaendfir6s.t7%and

secoofnitds hDaCrmcoomnipcocnoemntpLoDnCe,nrtesspoefcsteivlfe-liyn.dMucotraenocveera,rtehseigDnCificcoamntp,ownhenicthoaf cmcouutunatlfionrdaurcotuancde1M0.D1C%, aasnd

6.7s%hoowfnitisnDTCabcloem1,piosnnenumt LeDriCca, lrleyspabeoctuitve1l6y..4%Morfetohveesr,eltfh-iendDuCctaconmcepDoCnencotmofpomnuentut aLlDiCn, dwuhcitcahnce MDshCo, ualsdsbheotwaknenininTtaobalceco1u, nist.nAulmtheoruigcahlltyheabfoouurth16h.a4r%moonficthceomseplfo-ninedntuoctfasneclfe-iDndCucctoamncpeoannednthLeDC, whsiecchonshdohualrdmboentiackceonmipnotoneanctcouf nmtu. tAulatlhionudguhcttahnecefoaurrethnohtaarsmnoengilcigciobmlepaosnaellntthoefostehlef-rincodmucptoanecnetas,nd

theinseocrodnedr htoarfmacoilnitiactceotmhepoFnMePnMt ofmmauchtuinael imndoudcetlainngc,eaalrlethneosteashanremgloingicblceoams paollntehnetos tahreer ncoegmlepcotende.nts, in Tohrduesr, tthoefapchilaisteatientdhuectFaMncPeMcamn abcehaipnpermoxoidmealitnelgy, eaxllptrheessseedhaarsm: onic components are neglected. Thus,

the phase inductance can be apprLoaaximLDCateLlmy1 ceoxspree ssLemd2 coass:2e

Laa LDCLb-b LLmDC1cLoms1(cose)e+1L2m0o2 coLsm(2 c2ose2) e 120o

Lbb LDCLc-c LLmDC1cLoms1 (cose -e 112200oo)+Lm2Lcmos2 c2ose{21(20oe - 120o)}

Lcc LDCM-ab L mM1bccoMs(ca e +Mb1a 20Moc)b +MLacm2McoDCs{2(e + 120o)}

(4) (4)

where Lm1 and Lm2 Maraeb =theMbpce=ak Mvcaalu=esMobfa =theMcfbirs=t Manadc seMcoDnCd harmonic components of

self-inductance, respectively.

where Lm1 and Lm2 are the peak values of the first and second harmonic components of self-inductance, respectivTealbyl.e 1. Inductance characteristics in stator reference frame.

Items

Self-Inductance

Mutual Inductance

TableP1e.aIknVdualcutaen(cmeHch) araPcthearsisetiAcsnginlestatoPrearkefVeraelnucee(fmraHm)e. Phase Angle

DC component

28.711

-

-4.7168

-

First harmonic

2.9S0e2l2f-Inductance 2.9?

0.81M27utual Induct-a7n6c.5e?

SecIotenmd sharmonic Peak Va1l.u9e16(1mH) Phase6A? ngle Peak1V.3a5l3ue (mH) P-h5a1s.1e?Angle

Third harmonic

0.2063

9.3?

0.03

-72.7?

DCFocuormthphoanremntonic FirsFtifhtharhmaormniocnic Second harmonic

281.7.215122 2.900.027284 1.9161

2116.13-9..86??

-0.428.79168 0.01.4851427

1.353

-7113--.3.475??61- ..51

Third harmonic

0.2063

9.3

0.03

-72.7

TFhouusr,tthhheavromltoangiecequation1s.2o5f2th2e proposed m1a1c.h8ine in the stator0r.e2f8e9rence frame can71b.3ewritten as:

Fifth harmonic

0.0784

13.6

da 0.1454

-13.4

Thus,

the

voltage

equations

of

ua

thueb

Ra 0

pro0poRsbed

00m aiiabchindeddttbin

the

stator

reference

frame

ca(n5) be

written as:

ua

uc 0

Ra 0

0

Rc ic

dc

0 ia dt

da dt

where Ra, Rb, and Rc are theupbhase=resist0anceRs,b wh0ichare iebqua+l duedtdotbsymmetrical windings, thus(5)

termed

as

R.

The

power

absourcbed

by

the0wind0ingsRfcrom

thiec

power

sudppc ly

dt

can

be

expressed

as:

where Ra, Rb, and Rc are the phase resisPtiancueaisa, wubhibichucaicre equal due to symmetrical wind(i6n)gs,

thus termed as R. The power absorbed by the windings from the power supply can be expressed as:

Pi = uaia + ubib + ucic

(6)

Energies 2016, 9, 1078

7 of 15

Substituting Equations (3) and (5) into Equation (6), and neglecting mutual inductances to

simplify derivation, the electromagnetic torque of the proposed machine in stator reference frame can

be derived as:

Te = Tpm + Tr

(7)

in which:

Tpm = pr

ia

dpma de

+

ib

dpmb de

+

ic

dpmc de

(8)

Tr

=

pr 2

i2a

dLaa de

+

i2b

dLbb de

+

i2c

dLcc de

(9)

where Tpm is the PM torque, and Tr is called the reluctance torque, which is caused by the fluctuation of the phase self-inductance with rotor positions.

3.2. Abc-dq Transformation

The vector-control strategy is based on the synchronous rotor frame, which rotates at the synchronous velocity. In order to get the two-phase rotary d-q axes electromagnetic parameters, the traditional Park matrix P3s/2r as shown in Equation (10) can be used for abc-dq transformation.

P3s/2r =

2

3

cos(e ) - sin(e)

cos(e - 120o) - sin(e - 120o)

cos(e + 120o)

- sin(e + 120o)

(10)

1/2

1/2

1/2

Thus, the PM flux linkage in the d-q reference frame can be derived as:

pmd

pma

m

pmq

=

P3s/2r

pmb

=

0

(11)

pm0

pmc

0

It can be seen from Equation (11) that the PM flux linkage in d-axis pmd is equal to the peak value of phase PM flux linkage in the stator reference frame. The PM flux linkages in q-axis pmq and in 0-axis pm0 are equal to zero. To verify the aforementioned analysis, Figure 5b gives the PM flux linkage in the d-q reference frame, which is transformed from the three-phase PM flux linkage in the

stator reference frame, as shown in Figure 5a, obtained by using FEA. The average value of pmd, pmq, and pm0 are summarized in Table 2. It can be seen that the results calculated from the math model are consistent with the FEA.

Table 2. PM flux linkage in d-q reference frame. FEA: finite element analysis.

Items

pmd pmq pm0

Flux Linkage (Wb)

From FEA From Math Model

1.2106 -0.000079 -0.000173

1.2031 0 0

Then, the inductances in d-q axes frame can be described as follows:

Ld Ldq Ld0

Laa Mab Mac

Lqd

Lq

Lq0

=

P3s/2r

Mba

Lbb

Mbc P3-s/12r

(12)

L0d L0q L0

Mca Mcb Lcc

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